Scalar curvature and harmonic maps to $S^1$

Abstract

For a harmonic map $u:M^3\to S^1$ on a closed, oriented $3$--manifold, we establish the identity $$2\pi {\int }_{\theta \in S^1}\chi ({\Sigma }_{\theta })\ge \frac{1}{2}{\int }_{\theta \in S^1}{\int }_{{\Sigma }_{\theta }}(|du{|}^{-2}|Hess(u){|}^2+R_M)$$ relating the scalar curvature $R_M$ of $M$ to the average Euler characteristic of the level sets ${\Sigma }_{\theta }=u^{-1}\{\theta \}$. As our primary application, we extend the Kronheimer--Mrowka characterization of the Thurston norm on $H_2(M;\mathbb{Z})$ in terms of $\Vert R_M^{-}{\Vert }_{L^2}$ and the harmonic norm to any closed $3$--manifold containing no nonseparating spheres. Additional corollaries include the Bray--Brendle--Neves rigidity theorem for the systolic inequality $(\min R_M)sy{s}_2(M)\le 8\pi$, and the well--known result of Schoen and Yau that $T^3$ admits no metric of positive scalar curvature.

v2: minor edits--corrected statements of rigidity/splitting results; comments welcome